A389378 - OEIS

A389378

a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,k) * binomial(n+2*k-1,n-3*k).

1, 0, 0, 3, 20, 75, 231, 770, 2988, 11955, 45925, 171897, 647855, 2482753, 9583140, 36942528, 142195548, 548152335, 2118999795, 8207880490, 31824803745, 123488787719, 479622307285, 1864764751038, 7256958446775, 28263369668950, 110152636824087, 429590725414725

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OFFSET

0,4

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = [x^n] 1/(1 - x^3 / (1 - x)^5)^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1 - x^3 / (1 - x)^5) ). See A389351.

MATHEMATICA

Table[Sum[Binomial[n+k-1, k]Binomial[n+2*k-1, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 04 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\3, binomial(n+k-1, k)*binomial(n+2*k-1, n-3*k));